For
a continuous angle-valued map defined on a compact ANR
,
a field and any integer
, one proposes a refinement
of the Novikov–Betti
numbers of the pair
and a refinement
of the
Novikov homology of
,
where
denotes the integral degree one cohomology class represented
by . The
refinement
is a configuration of points, with multiplicity located in
identified to
, whose total
cardinality is the
Novikov–Betti number of the pair. The refinement
is a configuration of
submodules of the
Novikov homology whose direct sum is isomorphic to the Novikov homology and with the same
support as of
.
When
, the
configuration
is convertible into a configuration of mutually orthogonal closed Hilbert submodules of the
–homology of the
infinite cyclic cover of
defined by
, which
is an
–Hilbert
module. One discusses the properties of these configurations, namely robustness with
respect to continuous perturbation of the angle-values map and the Poincaré duality
and one derives some computational applications in topology. The main results
parallel the results for the case of real-valued map but with Novikov homology and
Novikov–Betti numbers replacing standard homology and standard Betti
numbers.
